Large scale phenomena in models of statistical mechanics

统计力学模型中的大规模现象

• 批准号: 1106850
• 负责人: Marek Biskup
• 金额: $ 12.76万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106850&HistoricalAwards=false
• 关键词: Large; scale; phenomena; models; statistical

The project addresses a variety of mathematical problems at the borderline of probability theory and statistical mechanics. The common feature of these is a relatively simple formulation, intriguing underlying phenomena and deep connections to physical disciplines. The first round of problems pertains to the Random Conductance Model. These include derivation of scaling limits of random walks in disordered media, study of spectral properties of random Laplacians, analysis of fluctuations of effective conductivity, detailed control of heat-kernel decay, etc. The second area proposed to investigate is that of Gradient Fields. These problems are ubiquitous in physical and applied sciences but their mathematical understanding is currently stalled because of the inability to overcome non-convexity of the underlying interaction. We offer specific methods and approaches how this obstacle may technically be overcome. The third class of problems addresses the area of Disordered Systems. The specific subjects of interest are random-field spin models and the dynamics governed by the parabolic Anderson model. Precise mathematical approaches are outlined that could lead to significant improvements in the mathematical understanding of these problems. A good many of listed problems are devised with the intention to provide training, and inclusion in research, of graduate students and postdocs who have interest in Probability Theory and Mathematical Physics.The tremendous success of hard sciences in accurate description, modeling and even forecasting complex natural phenomena derives, in large part, from their foundation in rigorous mathematics. In the context of solid-state physics and material sciences, the mathematical methods most commonly used are those of probability and/or statistics. This comes as no surprise as these disciplines have been designed precisely to deal with systems involving large numbers of individual constituents. The present project studies three specific classes of problems in probability theory of large systems whose origin is rooted in physics of materials. The most pertinent general question we ask is how the structural details of materials, and their various inherent irregularities, exactly express themselves in their macroscopic properties. As a rule, we seek an enveloping principle, or a "physical law", that extracts, in quantitative or qualitative form, the essential features from the specifics. It is hoped that developing the theoretical foundations of such systems will eventually lead to significant advances in engineering applications. There is also an immediate impact for mathematics itself: analysis of complex phenomena inevitably involves a number of separate sub-disciplines of mathematics and will thus lead to fruitful exchange of ideas among them. The project naturally offers an opportunity to include graduate students and postdocs into the research environment at a top research US university.

该项目解决了概率论和统计力学边界上的各种数学问题。它们的共同特点是相对简单的公式，有趣的潜在现象和与物理学科的深刻联系。第一轮的问题属于随机电导模型。这些包括推导的标度限制的随机游走在无序介质中，随机拉普拉斯算子的光谱特性的研究，有效电导率的波动分析，详细控制的热核衰变等。第二个领域提出调查的是梯度场。这些问题在物理和应用科学中普遍存在，但由于无法克服潜在相互作用的非凸性，目前对它们的数学理解停滞不前。我们提供具体的方法和途径，如何在技术上克服这一障碍。第三类问题涉及无序系统领域。感兴趣的具体课题是随机场自旋模型和抛物线安德森模型的动力学。精确的数学方法概述，可能会导致显着改善这些问题的数学理解。许多列出的问题都是为了培养对概率论和数学物理感兴趣的研究生和博士后而设计的。硬科学在精确描述、建模甚至预测复杂自然现象方面的巨大成功在很大程度上源于它们在严格数学中的基础。在固体物理学和材料科学的背景下，最常用的数学方法是概率和/或统计。这并不奇怪，因为这些学科的设计正是为了处理涉及大量个体成分的系统。本项目研究三个具体类别的问题，在概率论的大系统，其起源是植根于物理材料。我们要问的最相关的一般性问题是，材料的结构细节及其各种固有的不规则性如何在其宏观性质中准确地表达出来。一般来说，我们寻求一个包容性原则，或“物理定律”，以定量或定性的形式，从细节中提取基本特征。人们希望，发展这种系统的理论基础将最终导致工程应用的重大进展。这对数学本身也有直接的影响：对复杂现象的分析不可避免地涉及到数学的一些独立的子学科，因此将导致它们之间富有成效的思想交流。该项目自然提供了一个机会，将研究生和博士后纳入美国顶尖研究型大学的研究环境。